We consider the system 04arrow.

  Alphabet:

    and : [c * c] --> c 
    arrow : [t * t] --> t 
    lessthan : [t * t] --> c 

  Rules:

    lessthan(arrow(x, y), arrow(z, u)) => and(lessthan(z, x), lessthan(y, u)) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).

We thus obtain the following dependency pair problem (P_0, R_0, static, formative):

  Dependency Pairs P_0:

    0] lessthan#(arrow(X, Y), arrow(Z, U)) =#> lessthan#(Z, X)   
    1] lessthan#(arrow(X, Y), arrow(Z, U)) =#> lessthan#(Y, U)   

  Rules R_0:

    lessthan(arrow(X, Y), arrow(Z, U)) => and(lessthan(Z, X), lessthan(Y, U)) 

Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.

We consider the dependency pair problem (P_0, R_0, static, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  lessthan#(arrow(X, Y), arrow(Z, U)) >? lessthan#(Z, X) 
  lessthan#(arrow(X, Y), arrow(Z, U)) >? lessthan#(Y, U) 
  lessthan(arrow(X, Y), arrow(Z, U)) >= and(lessthan(Z, X), lessthan(Y, U)) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  and = \y0y1.0 
  arrow = \y0y1.3 + 2y0 + 2y1 
  lessthan = \y0y1.0 
  lessthan# = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[lessthan#(arrow(_x0, _x1), arrow(_x2, _x3))]] = 6 + 2x0 + 2x1 + 2x2 + 2x3 > x0 + x2 = [[lessthan#(_x2, _x0)]] 
  [[lessthan#(arrow(_x0, _x1), arrow(_x2, _x3))]] = 6 + 2x0 + 2x1 + 2x2 + 2x3 > x1 + x3 = [[lessthan#(_x1, _x3)]] 
  [[lessthan(arrow(_x0, _x1), arrow(_x2, _x3))]] = 0 >= 0 = [[and(lessthan(_x2, _x0), lessthan(_x1, _x3))]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_0, R_0) by ({}, R_0).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.